Solution manual for the physics of sound 3rd edition by berg

a Position: Location, relative to some standard, measured in centimeters, meters, miles, inches, etc.. f Force: The "push" or "pull" exerted on an object, measured in Newtons N.. g Weig

(a) Position: Location, relative to some standard, measured in centimeters,

meters, miles, inches, etc

(b) Time: Duration, measured in seconds (s), minutes, hours, days, etc

(c) Velocity: A speed in a particular direction, measured in meters per

second (m/s), centimeters/second (cm/s), miles/hour (mph), etc

(d) Acceleration: The rate at which the velocity or speed is changing,

measured in m/s2

(e) Mass: The quantity of matter, measured in grams (g) or kilograms (kg)

(f) Force: The "push" or "pull" exerted on an object, measured in Newtons

(N)

(g) Weight: The force due to gravity, measured in Newtons (N) or pounds

(English unit)

(h) Pressure: Force per unit area, measured in N/m2

(i) Density: Mass per unit volume of any substance, measured in g/cm3, kg/m3, etc

2 (a) A linear restoring force obeys the equation F = -kx, where F is the

restoring force, x is the position measured with respect to the resting position, and k is a constant that depends on the elastic medium The minus sign indicates that the force is a restoring force, tending to pull the

object back to its resting position "Linear" means that the force required

to hold the object a distance x from its equilibrium position is linearly proportional to the distance x, with k the constant of proportionality

(b) Simple harmonic motion (SHM) occurs when a system with a resting position (equilibrium) and a linear restoring force is displaced from that resting position and released

3 (a) Examples of motion that are periodic but not SHM:

• the ticking sound from a grandfather clock

• the flapping of a bird wing

• the earth rotating on its axis

• a superball bouncing without losing energy

(b) Examples of (nearly) SHM:

• the motion of a pendulum

• the motion of a mass vibrating up and down on a spring

• a sinusoidal oscillator signal

(c) The brightness of a flashing automobile directional turn signal light is periodic, but not simple harmonic motion

(a) Triangular wave, T = 5 ms, A = 2V

-2 -1 0 1 2

t ime ( ms)

(b) Square w ave, T = 10 ms, A = 3V

-4 -2 0 2 4

time (ms)

(c) Saw tooth w ave, T = 8m s, A = 1V

-1 -0.5 0 0.5 1

tim e (m s )

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Only (a) is damped harmonic motion, because the period does not change as the amplitude decreases In (b) both the amplitude and the period change

7

8 (a) In phase = moving together

(b) Out of phase = moving "opposite" to each other

(c) 90o phase difference = whatever one object does the other one does ¼ period later

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10 Resonance: When we have a system with a resting position and a linear

restoring force, it has a natural frequency - the frequency at which it will

oscillate if displaced from its equilibrium position and released If this

system is driven by a periodic force with the same natural frequency (or very

nearly that frequency), the oscillation builds up and the driver/system

combination is said to be in "resonance." Examples include (a) pushing a child on a swing, (b) playing another instrument near a piano with its damper pedal depressed so that certain piano strings begin to vibrate A sitar (South Asian string instrument) uses the "sympathetic vibrations" of strings not actually plucked

11 No solution

Mass position

-6 -4 -2 0 2 4 6

time (seconds)

13 No solution

(a) Tone becomes louder with time

-15.0 -10.0 -5.0 0.0 5.0 10.0 15.0

time (ms)

(b) Tone becomes lower in pitch with time

-15.0 -10.0 -5.0 0.0 5.0 10.0 15.0

time (ms)

(c) Tone becomes softer and higher in pitch

-15 -10 -5 0 5 10 15

Time (ms)

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(b) x oscillation 90o behind the y oscillation:

15 No solution

16 No solution; see Richard Berg and Dieter Brill, "Speed of Sound Using

Lissajous Figures," The Physics Teacher, October 2004

17 Whenever the length of the second pendulum is very close in length to the first pendulum it will cause the first pendulum to vibrate more The motion will then transfer back and forth between the two pendula

T = 1s: f = 1T = 1s = 1Hz 1

1

T = 0.2s: f = 1T = 0.2s = 5Hz 1

T = 10ms: f = 1T = 10ms = 1 0.01s = 1 Hz 1

T = 0.0002s: f = T = 1 0.0002s = 5000Hz = 5kHz 1

T = 20µs: f = T = 1 20µs = 1 0.000020s = 50,000 Hz = 50kHz 1

T = 0.5s: f = 1T = 0.5s = 2Hz 1

2

T = 0.1s: f = 1T = 0.1s = 10Hz 1

T = 20ms: f = 1T = 20ms = 1 0.02s = 50Hz 1

T = 0.001s: f = T = 1 0.001s = 1,000Hz = 1kHz 1

T = 40µs: f = T = 1 40µs = 1 0.000040s = 25,000 Hz = 25kHz 1

f = 0.5 Hz: T = 1f = 0.5Hz = 2s 1

3

f = 2 Hz: T = 1f = 2 Hz = 0.5s 1

f = 100 Hz: T = 1f = 100Hz = 0.01s = 10ms 1

f = 5000Hz: T = 1f = 5000 Hz = 0.0002 s = 0.2 ms = 200µs 1

f = 10,000 Hz:T = 1f = 10000Hz = 0.0001 s = 0.1ms = 100µs 1

f = 1 Hz: T = 1f = 1Hz = 1s 1

4

f = 20 Hz: T = 1f = 20 Hz = 0.05s = 50ms 1

f = 250 Hz: T = 1f = 250Hz = 0.004s = 4ms 1

f = 10,000Hz: T = 1f = 10000 Hz = 0.0001s = 0.1ms = 100µs 1

f = 50,000 Hz:T = 1f = 50000Hz = 0.00002s = 0.02ms = 20µs 1

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6

2 MHz = 2,000,000Hz

1000 MHz = 1,000,000,000Hz (= 1GHz)

f = 20 Hz: T = 1f = 20Hz = 0.05s = 50ms 1

f = 20kHz = 20,000Hz: T = 1f = 20000Hz = 0.00005s = 0.05ms = 50µs 1